As kneemo has discussed, in 2000 Gunaydin et al described $E_8$ in a nonlinear fashion, on a space of dimension $57$. Where do we get the number $57$ from path spaces? The $56$ dimensional $2 \times 2$ FTS matrix was a circulant of type $(1,27)$. A natural ternary analogue would be a $3 \times 3$ operator of circulant form $(1,9,9)$, which has total dimension $57$.

We can map the $9$ length $2$ qutrit paths into the $27$ path qutrit tetractys by concatenating on the left or right with an $X$, $Y$ or $Z$, giving a total of six maps. Note how the extra dimension ($57$ compared to $56$) essentially arises from the extra diagonal term. From a categorical perspective, this reflects the replacement of an abstract duality by triality. The length $2$ paths were associated to an associahedron secondary polytope, and the concatenations embed this into a higher dimensional polytope associated to the three qutrits. However, the noncommutative and nonassociative geometry will take us beyond mere $1$-ordinal polytopes.

8 years ago

This appears to make a number of things clear. This seems to lead to a way that we may break the symmetry of through the introduction of an event horizon, just as AdS_4 ~ SO(6,2) = SU(2,2) can be decomposed into AdS_2xR^2 by the “peeling off” of conformal SL(2,R)’s. Here the AdS is reduced by placing an black hole in the AdS_4, so that near the BH horizon the AdS symmetry group is broken or reduced. This might then be extended further, where the AdS in higher dimensions has been “Euclideanized,” or we might take the E_{n(n)}’s here as E_{m,2(m.2)} for n = 8 and E_{m,1(m,1)} for n = 7. The pealing off here can be see with the E_{8(8)} decomposed as E_{7(7)}xSL(2,R) as we are reducing this by “peeling off” SL(2,C) ~ SL(2,R)^2 conformal quantum groups.

ReplyDeleteYes, but here we will continue as we have for years and view all classical Lie symmetries, and their apparent breaking, as emergent feautures of a simpler categorical quantum gravity.

ReplyDeleteI will say more later about the E7(7)xSL(2) idea. What is remarkable is how closely connected it is to fundamental concepts in category theory.

I can't comment a whole lot on category theory, though I know some of Grothendieck's topos theory. The obvious question is what is E_{7(7)}xSL(2,R) categorically equivalent to?

ReplyDeleteThe peeling of sl(2,R) off leads to an 8-fold mass spectrum that is equivalent to Zamolodchikov's massive conformal theories. Zamolodchikov demonstrated the emergence of E_8 symmetry in a c = 1/2 CFT perturbed by a mirror operator, such as the wedge potential above. The Hamiltonian H_{1/2} = H^{0}_{1/2} + h∫ Vdz dz* (here * means bar), for V a Z_2 operator.. The wedge potential above exists in an Ising chain, and this system describes the spin density of a system at a critical temperature. At the critical temperature there is a splitting of masses, or “mass breakdown” similar to what happens in a Landau electron fluid, where the spectrum has a golden mean distribution --- the same as the root vectors of the E_8. This physics has been observed in condensed matter physics as seen in

R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, ”Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry ,” Science 8 January 2010: 177-180.

I would hazard to guess that a peeling off of SL(2,R)’s is a form of holographic symmetry breaking, the most elementary being AdS_4 to AdS_2xS^2. The boundary of AdS_4 is “infinity,” and at r = R there is the event horizon of a BTZ type of black hole. A particle is repelled by the AdS boundary, the spacetime being hyperbolic. The BH with a BPS gauge charge, say the U(1) charge of EM, say Q < 0, will repel a - charged particle pair and attract the + charge into it. Yet the – charge will also be repelled by the AdS boundary. Thus the vacuum state with +/- virtual loops exhibits a CFT symmetry breaking that is reflected by the corresponding breaking of the AdS_4 into AdS_2xS^2. So for higher dimensions there are additional “peeling off” of SL(2,R) as we approach the BH horizon.

Now if category theory comes into play here that might indeed be interesting. The AdS boundary and the BH horizon define the UV and IR ends in the renormalization group flow of the theory. A categorical theory result might have the effect of saying,” though the theory looks widely different at r = ∞ and at r = R, the two are categorically equivalent. Physically this would say that even with mass breakdowns and symmetry breaking the topological quantum numbers map continuously between the two ends.

So if you can illuminate more about how category theory plays a role here I would be most interested in that,

THX LC

Hi Lawrence. Well, there are several issues to discuss here. First, we need to separate the categorical bones of the nonlinear group actions (which should be fairly simple) from the full manifold geometry (ie. continua), the latter being heinously complicated from the categorical view. In fact, in my study of arithmetic quantum information I am a 'quantum constructivist' and have no intention of buying, for instance, the axiom of choice. Note that in a topos there is no unique notion of the real or complex numbers. And we need higher dimensional toposes.

ReplyDeleteBut eventually we do need to construct a continuum in order to recover, most notably, GR. In my opinion this will never happen without us tackling the fundamental axiomatic issues from the categorical point of view. So think of a continuum space as having a canonical description in terms of some new kind of prime spaces. Then we would need a large computer to properly describe a space like $E_{7(7)}$, but the physics will not require this exercise.

Secondly, the $r = \infty$ and $r = R$ relation would manifest itself in new physics, which is to say an observer dependent Machian cosmology (that we sometimes discuss) for which this duality is prior to any continuum notion of spacetime. So there are no mass breakdowns and symmetry breakings, because that picture belongs to an old theory that happens to be incapable of describing quantized mass etc.

This ended up being a bit long

ReplyDeleteA major aspect of what I have been doing involves Kleinian groups, which are discrete groups on manifolds. I break this out some below, but the AdS group modulo a discrete system, such as a permutation group, constructs a Kleinian structure. The actual physics is discrete, and there is a connection between this discrete structure and Heisenberg groups. The discrete structure defines light cones on discrete folations which mathematically are equivalent to Heisenberg groups. This is where things get into topos theory. The Hesenberg group is a system of 3x3 matrices, where the middle entry can be a symplectic element making this matrix 3x3xI_n, that is nonzero on the top right and diagonal. This defines Borel sets which are the foundation of Topos theory and … the rest is a lot of complicated mathematics that I will not get into right now. The upshot I think is that light cone structure and Heisenberg groups have a common “origin.”

Issues of the axiom of choice are not relevant to the actual physics. The physics is in a discrete system, where the continuum aspects of the theory might be said to “flap in the breeze.” This connects with Goyal’s ideas of discrete quantum mechanics, which has Topos-like structure.

One interesting question is how this is related to conformal SL(2,R) groups. The conformal group is the set of operators for which

[a, a*] = 1 + R, or [x, p] = i(1 + iR)

which can be embedded into the Heisenberg group by an appropriate extension on the symplectic elements. This seems to connect confomal structure with these Heisenberg groups. So the peeling off of conformal groups with these reductions AdS_n --> AdS_{n-2}xS^2 has some relationship that I as yet do not understand with AdS~CFT and two dimensional conformal groups on string world sheets. The Heisenberg groups emerge from a conformal completion of the AdS_n, and this structure seems to emerge by this peeling off of SL(2)’s

In the next post I include a discussion on Kleinian groups, which is something I wrote not long ago to someone else on this matter of Kleinian group structures with anti-de Sitter space.

The AdS spacetime on a patch is a Minkowski or Einstein spacetime metric. This means that the evolute of AdS from a spatial surface is an entire spacetime. So there is a loss of causality here. What is then required is a conformal completion of AdS. In doing so the Cauchy data on the AdS is defined on a conformal set of metrics. The boundary space ∂AdS_{n+1} is a Minkowski spacetime, or a spacetime E_n that is simply connected that with the AdS is such that (AdS_{n+1})UE_n is the conformal completion of AdS_{n+1} which exhibits a conformal completion under the discrete action of a Klienian group. For the Lorentzian group SO(2,n) there exists the discrete group SO(2,n,Z) which is a Mobius group. For a discrete subgroup Γ subset SO(2,n,Z) that obeys certain regular properties for accumulation points in the discrete set AdS_{n+1}/Γ is a conformal action of Γ on the sphere S_n. This is then a map which constructs an AdS ~ CFT correspondence.

ReplyDeleteThe quotient space AdS/ Γ is a Kleinian structure. The group SO(2,n) is a map from the unit ball B_{n+1}, with boundary ∂B_{n+1} = S_n, into R^{n+1}. The discrete group Γ acts as a conformal on the sphere S^n by the action of the Mobius transformation on S_n. The discrete set of maps on S^n has accumulation points on the limit sphere S^n_∞ are determined by the limit set g_i \in G for i --> ∞. This is denoted by Λ(G), G = O(2,n). The discontinuous set is then the complement of this or Ω(G) = S_n - Λ(G). The manifold Ω(G)/G is an orbifold. This means that the Mobius transformation on the limit sphere S^2_∞ is equivalent to the conformal transformation of N^{n+1} which is equivalent to the isometries of AdS_{n+1}. The Ω(Γ)∩E_n/Γ is then a Lorentzian manifold ∂AdS_{n+1), and a set of discrete points in E_n pertaining to spatial hyperbolids of equivalent data. In this way the data on any spatial surface of AdS_{n+1} is contained in this conformal completeness of AdS_{n+1}. This is equivalent to the discrete action of Γ on S_n.

Ah, great, your Heisenberg ideas sound a lot like what we are doing. That's why I have been so obsessed with $3 \times 3$ matrices for years. But I can't say I have read much of the AdS/CFT literature, because I suspect a lot of it is NOT topos oriented!

ReplyDeleteThe classical geometry that I prefer to think about is twistor geometry, meaning (for instance) the modern off shell non local techniques for scattering amplitudes. People working in this area certainly are inspired by AdS/CFT physics, but I don't know many who think so much about the categorical aspects of it.

Re conformal groups, I would turn everything into twistor geometry, where the dimension of the information space really does go up with particle number, like the category theory says it should. Moreover, if we need to we can conveniently think of spaces in terms of moduli spaces, which also go up in dimension in the same way. Since the categorical bones lie here, ordinary spacetime becomes a pale reflection of an infinite hierarchy of information spaces, and Minkowski space in particular appears for the minimal case of four legs. We say that the empty associahedron (given by the single leaf tree) is like a three particle tree and this is why we don't see three particle amplitudes.

ReplyDeleteI will get back to this tomorrow, but I agree with twistor space. This is particularly the case whith the E_6 realization --- what happens with the decomposition by stripping off conformal SL(2)'s.

ReplyDeleteMuch of what you are doing is to equate this or find an equivalency with this sort of geometry with symbolic strings. The number of letters in each of these systems is governed by the magic square system Baez sets out.

More later, it got a bit late here.